We know from the beginning of William Faulkner's "A Rose for Emily" that Miss Emily Grierson is an anachronism, a throwback to another time and way of thinking. The entire town is curious to go into her house when she dies because she is literally not like them, and the narrator's first description of her supports that constant friction and conflict:
Alive, Miss Emily had been a tradition, a duty, and a care; a sort of hereditary obligation upon the town....
Like her house, Miss Emily's thinking and actions are different than those of most of the town. There are a few older folks, men her father's age, who still think in the older southern ways like Miss Emily does; however, most of the town has grown more modern and they do not understand or appreciate Miss Emily's idiosyncrasies and obstinacies.
Miss Emily's trouble with the town began when
Colonel Sartoris, the mayor—he who fathered the edict that no Negro woman should appear on the streets without an apron—remitted her taxes, the dispensation dating from the death of her father on into perpetuity.
Of course the townspeople are not going to be happy that she does not have to pay taxes, so eventually they send her a notice of taxes due. Miss Emily stubbornly insists that she owes no taxes and simply refuses to pay them. This conflict reveals her stubborn refusal, or perhaps her inability, to change with the times.
When her father dies, she refuses to let anyone come and take her father's body for three days, another indication that Miss Emily is intent on clinging to the past and another source of conflict with modern society.
Another conflict Miss Emily has with the townspeople is never tacitly spoken about with her, but it centers around a smell emanating from her house. While the younger members of the town council want to address the problem with Miss Emily directly, the older members know this must not be done, for it would be perceived as an insult to tell a lady that she (or her house) smells. Instead they sneak over at midnight and spread lye around the foundation of the house. Again, Miss Emily's stubborn resistance to deal with life in any other way than she was taught years ago puts her in a position of conflict with the town.
In a kind of break from her old southern ways, Miss Emily creates quite a modern scandal in Jefferson when she begins having a blatant affair with an unmarried man. Even worse, Homer Barron is a Yankee and a carpetbagger, again setting Emily in opposition to the social and moral mores of the the town. What this reveals about her is that she intends to do whatever she pleases, regardless of the town's feeble attempts to correct her behavior.
It is interesting to note that, while Miss Emily is hopelessly lost in the mindset of the genteel southern woman (which is how her father raised her), she is also a stubborn and intractable woman who is intent on doing whatever she pleases. She does not think about or consider any consequences for her actions, and when anything of the sort is brought to her attention, she assiduously ignores it.
Both of these qualities in Miss Emily, her unwillingness to modernize her thinking and her consistent recalcitrance (stubbornness), are revealed through her conflicts with her fellow citizens.